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r = cosϕ, r =

sin (ϕ - π 4),

r = sinϕ, r =

cos(ϕ - π 4),

10.9. (-π 4 £ ϕ £ π 2).

10.10. (0 £ ϕ £ 3π 4).

10.11. r = 6cos3ϕ,

10.12. r = 1 2 + sinϕ.

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x4dx

7.27. ∫

7.28. ∫

(4 + x2 )

(4 − x2 )

3 2

4 + x2

x2dx

7.29.

∫

7.30. ∫

1 − x2

)

1 − x2

4 − x2

(

Задача 8. Вычислить площади фигур, ограниченных графиками

функций.

y = ( x − 2)3 ,

y = x

, y = 0,

9 − x2

8.1. y = 4x − 8.

8.2.

(0 ≤ x ≤ 3).

y = 4 − x2 ,

y = sin x cos2 x,

y = 0,

8.3. y = x2 − 2x.

8.4. (0 ≤ x ≤ π 2).

y = x2

, y = 0,

y =

, y = 0,

4 − x2

8.5. x = 0,

x = 1.

8.6.

(0 ≤ x ≤ 2).

y = cos x sin2 x,

y = 0,

y =

y = 0,

ex −1,

8.7. (0 ≤ x ≤ π 2).

8.8.

x = ln 2.

y =

y = 0,

8.10. y = arccos x,

y = 0,

8.9.

1 + ln x

x = 1,

x = e3 .

x = 0.

y = ( x + 1)2 ,

y = 2x − x2 + 3,

8.11.

8.12.

y2 = x + 1.

y = x2 − 4x + 3.

y = x

y = 0,

x = arccos y,

x = 0,

36 − x2

8.13. (0 ≤ x ≤ 6).

8.14. y = 0.

y = x2

y = 0,

y = arctg x,

y = 0,

8 − x2

8.15. x =

8.16. (0 ≤ x ≤ 2

y = x 4 - x2 , y = 0,

8.17.

x = ey -1, x = 0,

8.18.

y = ln 2.

(0 £ x £ 2).

8.19. y =

= 0,

8.20. y =

, y = 0,

1 +

1 + cos x

x =1.

x = π 2, x = -π 2.

x = ( y - 2)3 ,

y = cos5 x sin 2x, y = 0,

8.21.

x = 4 y - 8.

8.22. (0 £ x £ π 2).

8.23. y =

y = 0,

8.24. x = 4 - y2 ,

(x2 +1)2

x =1.

x = y2 - 2 y.

1 x

x =

x = 0,

8.26. y =

, y

= 0,

8.25.

1 + ln y

y =1, y = e3 .

x = 2, x =1.

y = x2 16 - x2 , y = 0,

8.27.

8.28.

x = 4 - y2 , x = 0,

(0 £ x £ 4).

y = 0,

y =1.

y = ( x -1)2 ,

y = x2 cos x,

y = 0,

8.29. y2 = x -1.

8.30. (0 £ x £ π 2).

Задача 9. Вычислить площади фигур, ограниченных линиями,

заданными уравнениями.

9.1.y = 22 sin3 t,

x= 2 ( x ³ 2).2 cos34 t,x

	x = 4	(t - sin t ),
		(1 - cost ),
9.3.
9.3.	y = 4

y = 4 (0 < x < 8π , y ³ 4).

9.2. y y

9.4. y x

=2 cost,

=22 sin t,

=2 ( y ³ 2).

=16cos3 t,

=2sin3 t,

=2 ( x ³ 2).

x = 2cos t,

9.5. y = 6sin t,
y = 3 ( y ³ 3).
	3	t,
x =16cos		t,

y = sin3 t,
9.7.

x = 63 (x ³ 63 ).

( )

x = 3 t - sin t ,


9.9. y = 3(1 - cost ),
	y = 3 (0 < x < 6π , y ³ 3).
	y = 3 (0 < x < 6π , y ³ 3).
	x = 6	(t - sin t ),
		(1 - cos t ),
9.11.
9.11.	y = 6

	y = 9 (0 < x <12π , y ³ 9).
						3	t,
	x = 32cos						t,

9.13. y = sin3 t,

	x = 4 ( x ³ 4).
	x = 6	(t - sin t ),
		(1 - cost ),
9.15.
9.15.	y = 6

	y = 6 (0 < x <12π , y ³ 6).
	x =10			(t - sin t ),
				(1 - cost ),
9.17.
9.17.	y =10

	y =15 (0 < x < 20π , y ³15).
								3
					2 cos			3	t,
	x = 2				2 cos				t,

			2 sin3 t,
9.19. y =			2 sin3 t,
			( x ³1).
	x =1		( x ³1).

x = 2

(t - sin t ),

(1 - cost ),

9.6. y = 2

y = 3 (0 < x < 4π , y ³ 3).

x = 6cost,

9.8. y = 2sin t,

y =

( y ³

2 cos

x = 8

2 sin3 t,

9.10. y

( x ³ 4).

x = 4

2 cost,

x = 2

= 3

9.12. y

2 sin t,

( y ³ 3).

y = 3

x = 3cos t,

= 8sin t,

9.14. y

y = 4 ( y ³ 4).

x = 8cos

= 4sin3 t,

9.16.

x = 33 (x ³ 33 ).

					3	t,
x = 6cos						t,

y = 4sin3 t,
9.18.				(x ³ 2				).
x = 2
x = 2		3					3

	2 cost,
x =	2 cost,

			2 sin t,
9.20. y = 4			2 sin t,
		( y ³ 4).
y = 4		( y ³ 4).

	x = t - sin t,
		- cost,
9.21. y =1		- cost,
	y =1 (0 < x < 2π , y ³1).
	x = 8(t - sin t ) ,
		(1 - cost ),
9.23.
9.23.	y = 8

	y =12 (0 < x <16π , y ³12).
						3	t,
	x = 24 cos						t,
9.25. y = 2sin3 t,
					(x ³ 9				).
	x = 9
	x = 9		3					3
	x = 2	(t - sin t ),
		(1 - cost ),
9.27.
9.27.	y = 2

	y = 2 (0 < x < 4π , y ³ 2).

				2 cost,
	x = 2			2 cost,

				2 sin t,
9.29. y = 5				2 sin t,
			( y ³ 5).
	y = 5		( y ³ 5).

x = 8cos3 t,

9.22. y = 8sin3 t,

x =1 ( x ³1).

x = 9cost,

9.24. y = 4sin t,

y = 2 ( y ³ 2).

x = 3cost,

9.26. y = 8sin t,

y = 4

( y ³ 4

2 cos

x = 4

9.28. y =

2 sin3 t,

( x ³ 2).

x = 2

x = 4

(t - sin t ),

(1 - cost ),

9.30.

y = 4

y = 6 (0 < x < 8π , y ³ 6).

Задача 10. Вычислить площади фигур, ограниченных линиями,

заданными в полярных координатах.


10.1. r =	4cos3ϕ,				r = 2	(r ³ 2).		10.2. r = cos 2ϕ.
r =			cosϕ,		r = sinϕ,
		3						10.4. r = 4sin 3ϕ, r = 2 (r ³ 2).
10.3.	(0			£	ϕ £ π	2).

r = 2cosϕ,					r = 2		sinϕ,
						3		10.6. r = sin 3ϕ.
10.5.	(0			£	ϕ £ π	2).

10.7. r = 6sin 3ϕ,					r = 3	(r ³ 3).		10.8. r = cos3ϕ.


10.13. r = cosϕ,			r = sinϕ,
(0 £ ϕ £ π 2).
10.15. r = cosϕ,			r = 2cosϕ.
10.17. r =1 +		cosϕ.
	2
10.19. r = (5 2)sinϕ,				r = (3 2)sinϕ.
10.21. r = (3 2)cosϕ,				r = (5 2)cosϕ.
10.23. r = sin 6ϕ.
10.25. r = cosϕ + sinϕ.
10.27. r = 2cos 6ϕ.
10.29. r = 3sinϕ,			r = 5sinϕ.

10.14. r = 6sinϕ, r = 4sinϕ.

10.16. r = sinϕ, r = 2sinϕ.

10.18. r = 12 + cosϕ.

10.20. r =1 + 2 sinϕ.

10.22. r = 4cos 4ϕ.

10.24. r = 2cosϕ, r = 3cosϕ.

10.26. r = 2sin 4ϕ.

10.28. r = cosϕ − sinϕ.

10.30. r = 2sinϕ, r = 4sinϕ.

Задача 11. Вычислить длины дуг кривых, заданных уравнениями в прямоугольной системе координат.

11.1. y = ln x,						3		£ x £		15.
11.2. y =	x2		-	ln x			, 1 £ x £ 2.

4		2

11.3. y = 1 - x2					+ arcsin x,							0 £ x £ 7 9.
11.4. y = ln		5							£ x £
				,				3			8.

		2x

11.5. y = − ln cos x,

0 ≤ x ≤ π 6.

11.6. y = ex + 6,

≤ x ≤ ln

15.

11.7. y = 2 + arcsin

1 4 ≤ x ≤ 1.

x − x2

11.8. y = ln

(

x2 −

)

2 ≤ x ≤ 3.

1 ,

11.9. y =

+ arccos x,

0 ≤ x ≤ 8 9.

1 − x2

(

)

0 ≤ x ≤ 1 4.

11.10. y = ln 1 − x2

11.11. y = 2 + ch x,

0 ≤ x ≤ 1.

11.12. y = 1 − ln cos x,

0 ≤ x ≤ π 6.

11.13. y = ex + 13,

≤ x ≤ ln

24.

11.14. y = − arccos

0 ≤ x ≤ 1 4.

x − x2

11.15. y = 2 − ex ,

≤ x ≤ ln

11.16. y = arcsin x −

0 ≤ x ≤ 15 16.

1 − x2

11.17. y = 1 − ln sin x,

π 3 ≤ x ≤ π 2.

11.18. y =

1 − ln

(

)

3 ≤ x ≤ 4.

−1 ,

11.19. y =

− arccos

+ 5,

1 9 ≤ x ≤ 1.

x − x2

11.20. y =

1 − ex − e− x

0 ≤ x ≤ 3.

11.21. y = ln sin x,

π 3 ≤ x ≤ π 2.

11.22. y = ln 7 − ln x,

≤ x ≤

11.23. y = ch x + 3,

0 ≤ x ≤ 1.

11.24. y = 1 + arcsin x −

0 ≤ x ≤ 3 4.

1 − x2

11.25. y = ln cos x + 2,

0 ≤ x ≤ π 6.

11.26. y = ex + 26,

≤ x ≤ ln

24.

11.27. y =

ex + e− x

+ 3,

0 ≤ x ≤ 2.

11.28. y = arccos

−

+ 4,

0 ≤ x ≤ 1 2.

x − x2

11.29. y =

ex + e− x

+ 3

0 ≤ x ≤ 2.

11.30. y = ex + e,

≤ x ≤ ln

15.

Задача 12.

Вычислить

длины дуг кривых, заданных

параметрическими уравнениями.

	x = 5(t − sin t ),

12.1.		(1 − cos t ),
12.1.	y = 5	(1 − cos t ),
		0 ≤ t ≤ π .
		0 ≤ t ≤ π .
	x = 4	(cost + t sin t ),

12.3.		(sin t − t cost ),
12.3.	y = 4	(sin t − t cost ),
		0 ≤ t ≤ 2π .
		0 ≤ t ≤ 2π .
		3

	x = 10cos t,

12.5. y = 10sin3 t,

	0 ≤ t ≤ π 2.
	x = 3(t − sin t ),

12.7.
12.7.	y = 3(1 − cost ),

	π ≤ t ≤ 2π .
	x = 3(cost + t sin t ),

12.9.
12.9.	y = 3(sin t − t cos t ),

0 ≤ t ≤ π 3.

	x = 3(2cost − cos 2t ),

12.2.
12.2.	y = 3(2sin t − sin 2t ),
						0 ≤ t ≤ 2π .
						0 ≤ t ≤ 2π .
				2
	x = (t			2	− 2)sin t + 2t cost,
	x = (t				− 2)sin t + 2t cost,
12.4.		(2 − t2 )cost + 2t sin t,
12.4.	y =	(2 − t2 )cost + 2t sin t,
									0 ≤ t ≤ π .
									0 ≤ t ≤ π .
		t		(cost + sin t ),
	x = e			(cost + sin t ),
12.6.
12.6.	y = et (cos t − sin t ),
						0 ≤ t ≤ π .
						0 ≤ t ≤ π .
		1					1
	x =		cost −						cos 2t,
	x =	2	cost −					4	cos 2t,
12.8.		1					1
12.8.		1					1
	y =			sin t −					sin 2t,
	y =	2		sin t −			4		sin 2t,
		2					4
		π 2 ≤ t ≤ 2π 3.
					2
	x = (t				2	− 2)sin t + 2t cost,
	x = (t					− 2)sin t + 2t cost,

12.10.		− t2 )cost + 2t sin t,
12.10.	y = (2	− t2 )cost + 2t sin t,

0 ≤ t ≤ π .

				3
	x = 6cos			3	t,
	x = 6cos				t,
		= 6sin3 t,
12.11. y		= 6sin3 t,

	0 ≤ t ≤ π 3.
	x = 2,5(t − sin t ),

12.13.		= 2,5(1 − cost ),
12.13.	y	= 2,5(1 − cost ),
			π 2 ≤ t ≤ π .
			π 2 ≤ t ≤ π .
	x = 6(cost + t sin t ),

12.15.		= 6(sin t − t cost ),
12.15.	y	= 6(sin t − t cost ),
					0 ≤ t ≤ π .
					0 ≤ t ≤ π .
	x = 8cos3 t,

		= 8sin3 t,
12.17. y		= 8sin3 t,

	0 ≤ t ≤ π 6.
	x = 4		(t − sin t ),

12.19.		= 4	(1 − cos t ),
12.19.	y	= 4	(1 − cos t ),

	π 2 ≤ t ≤ 2π 3.
	x = 8(cost + t sin t ),

12.21.		= 8	(sin t − t cost ),
12.21.	y	= 8	(sin t − t cost ),
			0 ≤ t ≤ π 4.
			0 ≤ t ≤ π 4.
				3
		= 4cos t,
	x	= 4cos t,
		= 4sin3 t,
12.23. y		= 4sin3 t,

	π 6 ≤ t ≤ π 4.
	x = 2		(t − sin t ),

12.25.		= 2	(1 − cos t ),
12.25.	y	= 2	(1 − cos t ),

0 ≤ t ≤ π 2.

		t	(cost + sin t ),
	x = e		(cost + sin t ),
12.12.
12.12.	y = et (cos t − sin t ),
				π 2 ≤ t ≤ π .
				π 2 ≤ t ≤ π .
	x = 3,5(2cos t − cos 2t ),

12.14.
12.14.	y = 3,5(2sin t − sin 2t ),
				0 ≤ t ≤ π 2.
				0 ≤ t ≤ π 2.
			2
	x = (t		2	− 2)sin t + 2t cost,
	x = (t			− 2)sin t + 2t cost,
12.16.
12.16.	y = (2 − t2 )cost + 2t sin t,
				0 ≤ t ≤ π 2.
				0 ≤ t ≤ π 2.
		t	(cost + sin t ),
	x = e		(cost + sin t ),
12.18.
12.18.	y = et (cos t − sin t ),
				0 ≤ t ≤ 2π .
				0 ≤ t ≤ 2π .
	x = 2(2cost − cos 2t ),

12.20.			(2sin t − sin 2t ),
12.20.	y = 2		(2sin t − sin 2t ),
				0 ≤ t ≤ π 3.
				0 ≤ t ≤ π 3.
			2
	x = (t		2	− 2)sin t + 2t cost,
	x = (t			− 2)sin t + 2t cost,
12.22.
12.22.	y = (2 − t2 )cost + 2t sin t,
				0 ≤ t ≤ 2π .
				0 ≤ t ≤ 2π .
		t	(cost + sin t ),
	x = e		(cost + sin t ),
12.24.
12.24.	y = et (cos t − sin t ),
				0 ≤ t ≤ 3π 2.
				0 ≤ t ≤ 3π 2.
	x = 4(2cost − cos 2t ),

12.26.			(2sin t − sin 2t ),
12.26.	y = 4		(2sin t − sin 2t ),

0 ≤ t ≤ π .

	x = 2	(cost + t sin t ),

12.27.		(sin t − t cost ),
12.27.	y = 2	(sin t − t cost ),

0 ≤ t ≤ π 2.

x = 2cos3 t,

12.29. y = 2sin3 t, 0 ≤ t ≤ π 4.

			2
	x = (t		2	− 2)sin t + 2t cost,
	x = (t			− 2)sin t + 2t cost,
12.28.
12.28.	y = (2 − t2 )cost + 2t sin t,
				0 ≤ t ≤ 3π .
				0 ≤ t ≤ 3π .
		t	(cost + sin t ),
	x = e		(cost + sin t ),
12.30.
12.30.	y = et (cos t − sin t ),

π 6 ≤ t ≤ π 4.

Задача 13. Вычислить длины дуг кривых, заданных уравнениями в полярных координатах.

13.1.ρ = 3e3ϕ 4 , − π 2 ≤ ϕ ≤ π 2.

13.2.ρ = 2e4ϕ 3 , − π 2 ≤ ϕ ≤ π 2.

13.3.ρ = 2 eϕ , − π 2 ≤ ϕ ≤ π 2.

13.4.ρ = 5e5ϕ 12 , − π 2 ≤ ϕ ≤ π 2.

13.5.ρ = 6e12ϕ 5 , − π 2 ≤ ϕ ≤ π 2.

13.6. ρ = 3e3ϕ 4 ,			0 ≤ ϕ ≤ π 3.
13.7. ρ = 4e4ϕ 3 ,			0 ≤ ϕ ≤ π 3.
13.8. ρ =		eϕ ,	0 ≤ ϕ ≤ π 3.
	2
13.9. ρ = 5e5ϕ 12 ,			0 ≤ ϕ ≤ π 3.

13.10. ρ = 12e12ϕ 5 , 0 ≤ ϕ ≤ π 3.

13.11.ρ = 1 − sinϕ, − π 2 ≤ ϕ ≤ −π 6.

13.12.ρ = 2(1 − cosϕ ), − π ≤ ϕ ≤ −π 2.

13.13. ρ = 3(1 + sinϕ ),	− π 6 ≤ ϕ ≤ 0.
13.14. ρ = 4(1 − sinϕ ),	0 ≤ ϕ ≤ π 6.
13.15. ρ = 5(1 − cosϕ ),	− π 3 ≤ ϕ ≤ 0.
13.16. ρ = 6(1 + sinϕ ),	− π 2 ≤ ϕ ≤ 0.

13.17.	ρ = 7(1 - sinϕ ),	- π 6 £ ϕ £ π 6.
13.18.	ρ = 8(1 - cosϕ ),	- 2π 3 £ ϕ £ 0.

13.19.ρ = 2ϕ, 0 ≤ ϕ ≤ 34.

13.20.ρ = 2ϕ, 0 ≤ ϕ ≤ 43.

13.21.ρ = 2ϕ, 0 ≤ ϕ ≤ 512.

13.22.ρ = 2ϕ, 0 ≤ ϕ ≤ 125.

13.23.ρ = 4ϕ, 0 ≤ ϕ ≤ 34.

13.24.ρ = 3ϕ, 0 ≤ ϕ ≤ 43.

13.25.ρ = 5ϕ, 0 ≤ ϕ ≤ 125.

13.26. ρ = 2cosϕ,						0 ≤ ϕ ≤ π 6.
13.27. ρ = 8cosϕ,						0 ≤ ϕ ≤ π 4.
13.28. ρ = 6cosϕ,						0 ≤ ϕ ≤ π 3.
13.29. ρ = 2sinϕ,						0 ≤ ϕ ≤ π 6.
13.30. ρ = 8sinϕ,						0 ≤ ϕ ≤ π 4.
	Задача 14. Вычислить объёмы тел, ограниченных поверхностями.
14.1.	x2	+ y2 =1,				z = y, z = 0 ( y ³ 0).

9
14.2. z = x2 + 4 y2 ,						z = 2.
14.3.	x2	+	y2	- z2 =1, z = 0, z = 3.

9		4
14.4.	x2	+	y2	-	z2	= -1, z =12.

9		4		36
14.5.	x2	+	y2	+	z2	=1, z =1, z = 0.

16		9		4
14.6. x2 + y2 = 9,						z = y, z = 0 ( y ³ 0).

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